PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

396 Practical MATLAB® Applications for Engineers

The script fi le square_time_frq is executed and the spectrums of the pulses for τ = 1, 0.5,

0.25, and 0.125 are shown in Figure 4.45.

Note that contracting f(t) in the time domain, from τ = 1 s to τ = 0.125 s (by 1/8) pro-

duces an expansion in the frequency domain by a factor of 8, with the magnitude scaled

by 1/8, verifying the property f(at) ↔ (1/a)F(w/a).

For example, for τ = 1 s, the estimated bandwidth is 1 rad/s, and the magnitude at w =

0 is 0.5. For τ = 0.125 s, the estimated bandwidth is 8 rad/s, and the magnitude at w = 0

is 0.06 (which fully agrees with the theoretical value 0.5/8 = 0.06).

Example 4.7

Let us analyze the periodic triangular function f(t) shown in Figure 4.46, defi ned over a

period of T = 1, given by the following trigonometric FS:

ft

n

nw t

n

()
 sin( )

11

1

^0



∑

where T = 1 and

w

(^0) T

2

2





The equation for f(t) is given by f(t) = −t + 1, over the range 0 ≤ t ≤ 1 (one period).

spectrum of a square wave with tau=1

spectrum of a square wave with tau=0.25

spectrum of a square wave with tau=0.5

spectrum of a square wave with tau=0.125

0.6

0.4

0.2

0

−0.2

−0.4

− 20 020 −^20020

− (^20020) − 20 0 20
w=nwo (rad per sec.) w=nwo (rad per sec.)
w=nwo (rad per sec.) w=nwo (rad per sec.)
0.3
0.2
0.1
−0.1
−0.2
0
Amplitude
Amplitude
0.1
0.05
0
−0.05
−0.1
0.15
0.1
0.05
−0.05
−0.1
0
Amplitude
Amplitude
FIGURE 4.45
Spectrum plots of Example 4.6.